PROJECT DATA # 1 1. 2. 3. 4. Henry Cavendish (1731-1810) measured density of the earth relative to the density of water [1]. The number of measurements is 29. The "true" value is 5.517. Cavendish's work led others to accurate values for the gravitational constant (G) and Earth's mass [2]: Newtonian constant of gravitation: G = 6.67384 x 10-11 m3 kg-1 s-2 Standard uncertainty: 0.00080 x 10-11 m3 kg-1 s-2 Relative standard uncertainty: 1.2 x 10-4 Describe the given original data set. Statistically analyze the data set and data distribution in the sample. In Cavendish's data, assume that the measured density of the earth has a normal distribution with mean and standard deviation , both unknown. Construct the 95% confidence interval for . Is the "true" value of the density of the earth in this interval? What level of confidence would be best to identify the confidence interval? Construct the 95% confidence interval for . Analyze the level of confidence and accuracy of measured value. Illustrate graphically the assumption that the underlying distribution is normal. Discuss how error in Cavendish's measured value affects uncertainty of the gravitation constant. Cavendish's measured data the ratio of the earth's weight to an equivalent volume of water (page 520 in [1]): 5. Data set: 5.5, 5.61, 4.88, 5.07, 5.26, 5.55, 5.36, 5.29, 5.58, 5.65, 5.57, 5.53, 5.62, 5.29, 5.44, 5.34, 5.79, 5.1, 5.27, 5.39, 5.42, 5.47, 5.63, 5.34, 5.46, 5.3, 5.75, 5.68, 5.85 Needs["PhysicalConstants`"] General::obspkg: _PhysicalConstants`_ is now obsolete. The legacy version being loaded may conflict with current Mathematica functionality. See the Compatibility Guide for updating information. EarthMass::shdw: Symbol _EarthMass_ appears in multiple contexts _{PhysicalConstants`,Global`}_; definitions in context _PhysicalConstants`_ may shadow or be shadowed by other definitions. 6. EarthMass 5.97421024 Kilogram References: [1] H. Cavendish, "Experiments to determine the Density of the Earth," Phil. Trans. Roy. Soc. London, vol. 88, p. 469-526, 1798. Available: http://www.jstor.org/stable/106988 [2] The NIST Reference on Constants, Units and Uncertainty: Newtonian constant of gravitation G [Online]. Available: http://www.physics.nist.gov/cgibin/cuu/Value?bg PROJECT DATA # 2 1. 2. 3. Albert Michelson (1852 1931) measured the speed of light [1] using optical interferometer [2] in 1879. The "true" value of the speed of light in air is 299 734.5 km/s. Describe the given original data set Statistically analyze the data set and data distribution in the sample. In Michelson's data, assume that the measured density of the earth has a normal distribution with mean and standard deviation , both unknown. Construct the 95% confidence interval for . Is the "true" value of the density of the earth in this interval? What level of confidence would be best to identify the confidence interval? Construct the 95% confidence interval for Analyze the level of confidence and accuracy of measured value Illustrate graphically the assumption that the underlying distribution is normal. Discuss how error in measured value affects uncertainty of the speed of light constant. Michelson's measured data of speed of light in air are tabulated in [1] (example page from [1]): 4. 5. Michelson's measured set of data (speed of light in km/s): 299850,299740,299900,300070,299930,299850,299950,299980,299980,299880,3 00000,299980,299930,299650,299760,299810,300000,300000,299960,299960,29 9960,299940,299960,299940,299880,299800,299850,299880,299900,299840,299 830,299790,299810,299880,299880,299830,299800,299790,299760,299800,2998 80,299880,299880,299860,299720,299720,299620,299860,299970,299950,29988 0,299910,299850,299870,299840,299840,299850,299840,299840,299840,299890 ,299810,299810,299820,299800,299770,299760,299740,299750,299760,299910, 299920,299890,299860,299880,299720,299840,299850,299850,299780,299890,2 99840,299780,299810,299760,299810,299790,299810,299820,299850,299870,29 9870,299810,299740,299810,299940,299950,299800,299810,299870 References: [1] A. A. Michelson, "Experimental determination of the velocity of light made at the United States Naval Academy, Annapolis," Astronomical Papers, vol. 1, p. 109-145, 1882. Available: http://www.gutenberg.org/files/11753/11753-h/11753h.htm [2] Wikipedia, Michelson interferometer [Online]. Available: http://en.wikipedia.org/wiki/Michelson_interferometer PROJECT DATA # 3 1. James Short (1710 1768) a Scottish mathematician, optician and telescope maker measured the parallax of the sun (in seconds of a degree) originally published in [1], based on the 1761 transit of Venus [2]. 2. The parallax of the sun is the angle subtended by the earth, as seen from the surface of the sun. The number of data points in the measured data set is 53 and the "true" value is 8.798. Describe the given original data set Statistically analyze the data set and data distribution in the sample. In Short's data, assume that measured the parallax of the sun has a normal distribution with mean and standard deviation , both unknown. Construct the 95% confidence interval for . Is the "true" value of the density of the earth in this interval? What level of confidence would be best to identify the confidence interval? Construct the 95% confidence interval for Analyze the level of confidence and accuracy of measured value. Illustrate graphically the assumption that the underlying distribution is normal. Discuss how error in measured value affects uncertainty of the parallax of the sun. 3. Measured data the parallax of the sun in seconds of a degree (page 310 in [1]): 4. Data set: 8.5, 8.5, 7.33, 8.64, 9.27, 9.06, 9.25, 9.09, 8.5, 8.06, 8.43, 8.44, 8.14, 7.68, 10.34, 8.07, 8.36, 9.71, 8.65, 8.35, 8.71, 8.31, 8.36, 8.58, 7.8, 7.71, 8.3, 9.71, 8.5, 8.28, 9.87, 8.86, 5.76, 8.44, 8.23, 8.5, 8.8, 8.4, 8.82, 9.02, 10.57, 9.11, 8.66, 8.34, 8.6, 7.99, 8.58, 8.34, 9.64, 8.34, 8.55, 9.54, 9.07 References: [1] J. Short, "Second Paper concerning the Parallax of the Sun Determined from the Observations of the Late Transit of Venus, in Which This Subject is Treated of More at Length, and the Quantity of the Parallax More Fully Ascertained", Phil. Trans., vol. 53, p. 300-345, 1763. [2] Wikipedia, Transit of Venus [Online]. Available: http://en.wikipedia.org/wiki/Transit_of_Venus 5